Economic growth and economic development 695
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Introduction to Modern Economic Growth
15.4. Directed Technological Change with Knowledge Spillovers
We now consider the directed technological change model of the previous section
with a different specification of the innovation possibilities frontier. This is not only
useful to show that the results can be generalized, but also enables us to understand
the conditions leading to the strong bias result in Proposition 15.4 better.
The lab equipment specification of the innovation possibilities frontier is special
in one respect: it does not allow for state dependence. State dependence refers to
the phenomenon in which the path of past innovations affects the relative costs of
different types of innovations. The lab equipment specification implied that R&D
spending always leads to the same increase in the number of L-complementary and
H-complementary machines. We will now introduce a specification with knowledge
spillovers, which allows for state dependence. Recall that, as discussed in Section
13.2 in Chapter 13, when there are scarce factors used for R&D, then growth cannot
be sustained by continuously increasing the amount of these factors allocated to
R&D. Therefore, in order to achieve sustained growth, these factors need to become
more and more productive over time, because of spillovers from past research. Here
for simplicity, let us assume that R&D is carried out by scientists and that there
is a constant supply of scientists equal to S (Exercise 15.18 shows that the results
are identical when workers can be employed in the R&D sector). With only one
sector, the analysis in Section 13.2 in Chapter 13 indicates that sustained endogenous
˙
growth requires N/N
to be proportional to S. With two sectors, instead, there
is a variety of specifications with different degrees of state dependence, because
productivity in each sector can depend on the state of knowledge in both sectors.
A flexible formulation is the following:
(15.32)
N˙ L (t) = η L NL (t)(1+δ)/2 NH (t)(1−δ)/2 SL (t) and N˙ H (t) = η H NL (t)(1−δ)/2 NH (t)(1+δ)/2 SH (t) ,
where δ ≤ 1, and SL (t) is the number of scientists working to produce L-complementary
machines, while SH (t) denotes the number of scientists working on H-complementary
machines. Clearly, market clearing for scientists requires that
(15.33)
SL (t) + SH (t) ≤ S.
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